Question

One more Integration

Answer 1

\[\cos ^{3}(\frac{ x }{ 3})dx\]

Answer 2

\[\cos(x/3)-\cos(x/3)\sin ^{2}(x/3) dx\]

Answer 3

As far as I have got.

Answer 4

Actually\[\sin(x/3)- \int\limits \cos(x/3)\sin ^{2}(x/3)dx\]

Answer 5

Well, that's good so far. So, the cos(x/3) part can be integrated as it is. The 2nd portion requires just a simple u-substitution.

Answer 6

And its not sin(x/3) for the first part you integrated, though.

Answer 7

3sin(x/3)

Answer 8

There ya go. Now your second integral just needs a u-substitution to solve.

Answer 9

3sin(x/3)-9(sin(x/3)^3 +C

Answer 10

Maybe 1/9 not 9

Answer 11

Well, if u = sin(x/3)
du = (1/3)cos(x/3)dx, and dx = 3du/(cos(x/3))
So if I replace sin(x/3) with u and dx with 3du/(cos(x/3)), Ill be left with:
\[\int\limits_{}^{}\frac{ \cos(x/3)*u^{2}(3 du)}{ \cos(x/3) } \implies 3*\int\limits_{}^{}u^{2}du\]
Did you get the same result after substitution? If you integrate what I have, your answer should come out correct :)

Answer 12

3sin(x/3)-sin^3(x/3)+C

Answer 13

You've been a great help..Thanks

Answer 14

LaTeX Note: If you just \left( \right ) paraentheses in pairs, you will get a better result than just ().